Root Locations of an Entire Polytope of Polynomials: It Suffices to Check the Edges

The presence of uncertain parameters in either a state space or frequency domain description of a linear, time-invariant system manifests itself as variations in the coefficients of the characteristic polynomial. If the family of all such polynomials is polytopic in coefficient space, we´ll show that the root locations of the entire family can be completely determined by examining only the roots of the polynomials contained in the exposed edges of the polytope. These results are computationally feasible and this crterion goes beyond the presently available stability tests for uncertain systems by being less conservative in all cases and by explicitly determining all root locations. Equally important is the fact that these results are also applicable to discrete-time systems